Valuative uniqueness diagram

Let $f:X\to S$ be a morphism of schemes. A valuative diagram for $f$ consists of a valuation ring $R\subseteq K$ with fraction field $K$ (Valuation rings, The field of fractions $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$ of an integral domain) together with a morphism $\mathrm{Spec}K\to X$ and a morphism $\mathrm{Spec}R\to S$ whose composite with $\mathrm{Spec}K\to \mathrm{Spec}R$ is $f\circ (\mathrm{Spec}K\to X)$; that is, the square $\begin{array}{ccc}\mathrm{Spec}K& ⟶& X\\ \downarrow & & \downarrow f\\ \mathrm{Spec}R& ⟶& S\end{array}$ commutes. Call $\mathrm{Spec}K\to X$ the generic map of the diagram. A lift of the diagram is a morphism $\mathrm{Spec}R\to X$ making both triangles commute.

The morphism $f$ satisfies the uniqueness part of the valuative criterion if every valuative diagram for $f$ has at most one lift, and the existence part if every valuative diagram for $f$ has at least one lift. The two quantifiers are separate: uniqueness asserts nothing when no lift exists, and existence asserts nothing about the number of lifts.

The criterion is stated for all valuation rings, all fraction fields, all specializations of the closed point, and all prescribed extensions of residue fields implicit in the choice of the generic map; nothing restricts $R$ to discrete valuation rings, to Noetherian rings or to finite-type situations. Separatedness of $f$ is a property of the morphism alone, whereas a valuative diagram involves chosen maps from the spectra of $R$ and $K$; the two are compared by the lemmas and the theorem following on this page.